Exercise 9.4.6.25. In the special case $\operatorname{\mathcal{C}}= \Delta ^2$ and $\operatorname{\mathcal{C}}_0 = \Lambda ^{2}_{1}$, Proposition 9.4.6.24 reduces to the assertion that every inner fibration $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \Lambda ^{2}_{1}$ fits into a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r]^-{F} \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Lambda ^{2}_{1} \ar [r] & \Delta ^2, } \]
where $\operatorname{\mathcal{E}}$ is an $\infty $-category and $F$ is a categorical equivalence of simplicial sets (see Corollary 9.4.6.16). Use the small object argument to give a more direct proof of this statement.